Reflection from Layered Surfaces Due to Subsurface Scattering

Reﬂection from Layered Surfaces due to Subsurface Scattering Pat Hanrahan Wolfgang Krueger

Department of Computer Science Department of Scientiﬁc Visualization Princeton University German National Research Center for Computer Science

Abstract of incidence. Diffuse reflection is qualitatively explained as due to subsurface scattering [18]: Light enters the material, is absorbed The reflection of light from most materials consists of two ma- and scattered, and eventually exits the material. In the process of jor terms: the specular and the diffuse. Specular reflection may this subsurface interaction, light at different wavelengths is differ- be modeled from first principles by considering a rough surface entially absorbed and scattered, and hence is filtered accounting consisting of perfect reflectors, or micro-facets. Diffuse reflection for the color of the material. Moreover, in the limit as the light ray is generally considered to result from multiple scattering either is scattered multiple times, it becomes isotropic, and hence the di- from a rough surface or from within a layer near the surface. Ac- rection in which it leaves the material is essentially random. This counting for diffuse reflection by Lambert’s Cosine Law, as is qualitative explanation accounts for both the directional and col- universally done in computer graphics, is not a physical theory ormetric properties of diffuse materials. This explanation is also based on first principles. motivated by an early proof that there cannot exist a micro-facet This paper presents a model for subsurface scattering in layered distribution that causes equal reflection in all outgoing directions surfaces in terms of one-dimensional linear transport theory. We independent of the incoming direction [10]. derive explicit formulas for backscattering and transmission that The above model of diffuse reflection is qualitative and not can be directly incorporated in most rendering systems, and a gen- very satisfying because it does not refer to any physical param- eral Monte Carlo method that is easily added to a ray tracer. This eter of the material. Furthermore, there is no freedom to adjust model is particularly appropriate for common layered materials coefficients to account for subtle variations in reflection from dif- appearing in nature, such as biological tissues (e.g. skin, leaves, ferent materials. However, it does contain the essential insight: etc.) or inorganic materials (e.g. snow, sand, paint, varnished or an important component of reflection can arise from subsurface dusty surfaces). As an application of the model, we simulate the scattering. In this paper, we present a model of reflection of light appearance of a face and a cluster of leaves from experimental due to subsurface scattering in layered materials suitable for com- data describing their layer properties. puter graphics. The only other work in computer graphics to take CR Categories and Subject Descriptors: I.3.7 [Computer this approach is due to Blinn, who in a very early paper presented Graphics]: Three-Dimensional Graphics and Realism. a model for the reflection and transmission of light through thin Additional Key Words and Phrases: Reflection models, integral clouds of particles in order to model the rings of Saturn[2]. Our equations, Monte Carlo. model differs from Blinn’s in that it is based on one-dimensional linear transport theory—a simplification of the general volume rendering equation [19]— and hence is considerably more general 1 Motivation and powerful. Of course, Blinn was certainly aware of the trans- An important goal of image synthesis research is to develop a port theory approach, but chose to present his model in a simpler comprehensive shading model suitable for a wide range of ma- way based on probabilistic arguments. terials. Recent research has concentrated on developing a model In our model the relative contributions of surface and subsur- of specular reflection from rough surfaces from first principles. face reflection are very sensitive to the Fresnel effect (which Blinn In particular, the micro-facet model first proposed by Bouguer did not consider). This is particularly important in biological tis- in 1759 [4], and developed further by Beckmann[1], Torrance & sues which, because cells contain large quantities of water, are Sparrow[26], and others, has been applied to computer graphics translucent. A further prediction of the theory is that the sub- by Blinn [2] and Cook & Torrance[8]. A still more comprehen- surface reflectance term is not necessarily isotropic, but varies in sive version of the model was recently proposed by He et al[12]. different directions. This arises because the subsurface scattering These models have also been extended to handle anisotropic mi- by particles is predominantly in the forward direction. In fact, it crofacets distributions[24, 5] and multiple scattering from complex has long been known experimentally that very few materials are microscale geometries[28]. ideal diffuse reflectors (for a nice survey of experiments pertaining Another important component of surface reflection is, however, to this question, see [18]). diffuse reflection. Diffuse reflection in computer graphics has al- We formulate the model in the currently emerging standard most universally been modeled by Lambert’s Cosine Law. This terminology for describing illumination in computer graphics [16, law states that the exiting radiance is isotropic, and proportional 11]. We also discuss efficient methods for implementation within to the surface irradiance, which for a light ray impinging on the the context of standard rendering techniques. We also describe surface from a given direction depends on the cosine of the angle how to construct materials with multiple thin layers. Finally, we apply the model to two examples: skin and leaves. For these examples, we build on experimental data collected in the last few PermissionPermission to to copy copy without without fee fee all all or or part part of of this this material material is is granted granted years, and provide pointers to the relevant literature. pprovidedrovided that that the the copies copies are are not not made made or or distributed distributed for for direct direct Another goal of this paper is to point out the large amount of ccommercialommercial advantage, advantage, the the ACM ACM copyright copyright notice notice and and the the title title of of the the recent work in the applied physics community in the application ppublicationublication and and its its date date appear, appear, and and notice notice is is given given that that copying copying is is by by of linear transport theory to modeling appearance. ppermissionermission of of the the Association Association for for Computing Computing Machinery. Machinery. 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1.0 Li Lr,s Lr,v θ 0.8 i θr θr layer1 !!! !!!!!!!!! !!! 0.6 !!!!!! !!!!!!!!! !!!!!! !!!!!!!!!!!!!!!!!!!!! !!!!!! θ d!!!!!!layer2!!!!!!!!!!!! ’ !!!!!!!!! !!! !!!!!!!!!!!! !!!!!!!!!!!!!!! 0.4 !!!!!!!!!!!!!!!!!!!!! θ !!!!!!!!!!!! Amplitude !!! !!!!!!!!! !!!!!! 666666666666666666666666666!!! !!! 0.2 666666666666666666666666666layer3 666666666666666666666666666 666666666666666666666666666θ θ 0.0 666666666666666666666666666t t Lri Lt,v 0 10 20 30 40 50 60 70 80 90 z Angle of incidence Figure 1: The geometry of scattering from a layered surface Figure 2: Fresnel transmission and reflection coefficients for a ray leaving air (n = 1.0) and entering water (n = 1.33). (θi, φi) Angles of incidence (incoming) (θr, φr) Angles of reflection (outgoing) Lt,v - transmitted radiance due to volume or subsurface scat- (θt, φt) Angles of transmission tering L(z; θ, φ) Radiance [W / (m2 sr)] Li Incident (incoming) radiance The bidirectional reflection-distribution function (BRDF) is de- Lr Reflected (outgoing) radiance fined to the differential reflected radiance in the outgoing direction Lt Transmitted radiance per differential incident irradiance in the incoming direction [23]. L+ forward-scattered radiance L backward-scattered radiance L (θ , φ ) − r r r fr(θi, φi; θr, φr) BRDF fr(θi, φi; θr, θr) ≡ Li(θi, φi) cos θidωi ft(θi, φi; θt, φt) BTDF fr,s(θi, φi; θr, φr) Surface or boundary BRDF The bidirectional transmission-distribution function (BTDF) has a ft,s(θi, φi; θt, φt) Surface or boundary BTDF similar definition: fr,v(θi, φi; θr, φr) Volume or subsurface BRDF ft,v(θi, φi; θt, φt) Volume or subsurface BTDF Lt(θt, φt) ft(θi, φi; θt, θt) n Index of refraction ≡ Li(θi, φi) cos θidωi 1 σs(z; λ) Scattering cross section [mm− ] 1 Since we have separated the reflected and transmitted light into σa(z; λ) Absorption cross section [mm− ] 1 two components, the BRDF and BTDF also have two components. σt(z; λ) Total cross section (σt = σa + σs) [mm− ] W Albedo (W = σs ) σt fr = fr,s + fr,v d Layer thickness [mm] p(z; θ, φ; θ0, φ0; λ) Scattering phase function ((θ0, φ0) to (θ, φ)) ft = fri + ft,v If we assume a planar surface, then the radiance reflected from Table 1: Nomenclature and transmitted across the plane is given by the classic Fresnel coefficients. 2 Reflection and Transmission due to Layered 12 Surfaces Lr(θr, φr) = R (ni, nt; θi, φi θr, φr)Li(θi, φi) 12 → Lt(θt, φt) = T (ni, nt; θi, φi θt, φt)Li(θi, φi) As a starting point we will assume that the reflected radiance Lr → from a surface has two components. One component arises due to where surface reflectance, the other component due to subsurface volume R12(n , n ; θ , φ θ , φ ) = R(n , n , cos θ , cos θ ) scattering. (The notation used in this paper is collected in Table 1 i t i i r r i t i t → 2 2 and shown diagramatically in Figure 1.) 12 nt nt T (ni, nt; θi, φi θt, φt) = 2 T = 2 (1 R) → ni ni − Lr(θr, φr) = Lr,s(θr, φr) + Lr,v (θr, φr) where: where R and T are the Fresnel reflection formulae and are de- L - reflected radiance due to surface scattering r,s scribed in the standard texts (e.g. Ishimura[14]) and θt is the Lr,v - reflected radiance due to volume or subsurface scattering angle of transmission. Besides returning the amount of reflection The models developed in this paper also predict the transmis- and transmission across the boundary, the functions R12 and T 12, sion through a layered surface. This is useful both for materials as a side effect, compute the reflected and refracted angles from the made of multiple layers, as well as the transmission through thin Reflection Law (θr = θi) and Snell’s Law (ni sin θi = nt sin θt). 2 translucent surfaces when they are back illuminated. The transmit- Note also the factor of (nt/ni) ) in the transmitted coefficient of ted radiance has two components. The first component is called the above formula; this arises due to the change in differential the reduced intensity; this is the amount of incident light trans- solid angle under refraction and is discussed in Ishimura[pp. 154- mitted through the layer without scattering inside the layer, but 155]. Plots of the Fresnel functions for the boundary between air accounting for absorption. The second is due to scattering in the and water are shown in Figure 2. volume. In our model of reflection, the relative contributions of the sur- Lt(θt, φt) = Lri(θt, φt) + Lt,v(θt, φt) face and subsurface terms are modulated by the Fresnel coeffi- where: cients. Lri - reduced intensity fr = Rfr,s + Tfr,v = Rfr,s + (1 R)fr,v −

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Thus, an immediate prediction of the model is that reflection due to subsurface scattering is high when Fresnel reflection is low, since more light enters the surface layer. Notice in Figure 2 that the percentage of transmission is very high for a quite wide range of Figure 3: Henyey-Greenstein phase function for g = .3 and − angles of incidence. Thus, the reflectance properties of materials g = .6. impregnated with water or oil (dielectrics with low indices of refraction) are dominated by subsurface reflectance components at form and orientation of the suspended particles, the dielectric near perpendicular angles of incidence, and surface components properties of the particles, and the wavelength of the incident at glancing angles of incidence. light. The scattering of light from particles small compared to the wavelength of light is given by the Rayleigh scatter- Actually, light returning from the subsurface layers must refract ing formula, and the scattering due to dielectric spheres of across the boundary again. Thus, it will be attenuated by yet an- different radii by the Mie formula. other Fresnel transmission factor. Recall that if light returns from a media with a higher index of refraction, then total internal reflec- However, most materials contain distributions of particles tion may occur. All light with an incident angle greater than the of many different sizes, so simple single particle phase func- 1 tions are not applicable. For this reason, we describe the ma- critical angle (θc = sin− ni/nt) will not be transmitted across the boundary. By assuming an isotropic distribution of returning light, terial phase function with the empirical formula, the Henyey- we can compute the percentage that will be transmitted and hence Greenstein formula[13]. considered reflected. This sets an upper bound on the subsurface 1 1 g2 2 pHG(cos j) = − reflectance of 1 (ni/nt) (remember, nt > ni). For example, 4π (1 + g2 2g cos j)3/2 − − for an air-water boundary, the maximum subsurface reflectance is where j is the angle between the incoming and the outgoing approximately .44. direction (if the phase function depends only on this angle the scattering is symmetric about the incident direction). 3 Description of Materials The Henyey-Greenstein formula depends on a single param- eter g, the mean cosine of the scattered light. The Henyey- The aim of this work is to simulate the appearance of natural ma- Greenstein phase function for different values of g is shown terials such as human skin, plant leaves, snow, sand, paint, etc. in Figure 3. Note that if g = 0 the scattering is isotropic, The surface of these materials is comprised of one or more layers whereas positive g indicates predominantly forward scatter- of material composed of a mixture of randomly distributed parti- ing and negative g indicates predominantly backward scat- cles or inhomogeneities embedded in a translucent media. Particle tering. distributions can also exist, in which case the properties are the material are given by the product of each particle’s properties In the model employed in this paper, material properties are times the number of particles per unit volume. described macroscopically as averages over the underlying microscopic material property definitions. If the material is made The layers of such materials can be described by a set of macro- of several components, the resulting properties of the composite scopic parameters as shown in the following table. Measurements materials can be computed by simple summation. of these properties have been made for a large variety of natural n materials. σa = wi σa,i Symbol Property i=1 n index of refraction Xn 1 σa [mm− ] absorption cross section σs p(cos j, g) = wi σs,i p(cos j, gi) 1 σs [mm− ] scattering cross section i=1 d [mm] depth or thickness X p(cos j) scattering phase function and so on. Here wi is the volume fraction of the volume occupied g mean cosine of phase function by material i. Another very important property of real materials is that the Index of Refraction properties randomly vary or fluctuate. Such fluctuations cause • The materials considered are dielectrics where n is on the variation in the appearance of natural surfaces. This type of fluc- order of the index of refraction of water (1.33). tuation is easy to model with a random noise function or a texture map. Absorption and scattering cross section Optical propagation in random media has been studied in a va- • The intensity of the backscattered and transmitted light de- riety of applications, including blood oximetry, skin photometry, pends on the absorption and scattering properties of the mate- plant physiology, remote sensing for canopies and snow, the paint rial. The cross section may be interpreted as the probability and paper industry, and oceanic and atmospheric propagation. For per unit length of an interaction of a particular type. The many examples the macroscopic parameters have been measured total scattering cross section σ = σ + σ . The mean free t a s across many frequency bands. A major attempt of our work is the path is equal to the reciprocal of the total cross section. An simulation of the appearance of natural surfaces by using mea- important quantity is the albedo, which equals W = σ /σ . s t sured parameters to be inserted into the subsurface reflection and If the albedo is close to 1, the scattering cross section is transmission formulas. This approach is similar to the attempt of much greater than the absorption cross section, whereas if Cook & Torrance [8] to simulate the appearance of metallic sur- the albedo is close to 0, absorption is much more likely than faces by using appropriate values for the refractive index and the scattering. roughness parameters. Scattering phase function • The phase function, p(~x; θ, φ; θ0, φ0) represents the direc- 4 Light Transport Equations tional scattering from (θ0, φ0) to (θ, φ) of the light incident onto a particle. This function depends on the nature of the Linear transport theory is a heuristic description of the propagation scattering medium. The form of p is affected by the size, of light in materials. Transport theory is an approximation to elec-

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tromagnetic scattering theory, and hence cannot predict diffrac- This simply states that the forward component of radiance entering tion, interference or quantum effects. In particular, the specular the volume at the boundary is due to light transmitted across the reflection of light from rough surfaces whose height variation is surface. If we assume a planar surface and parallel incident rays, comparable in size to the wavelength of incident light requires then ft,s equals the Fresnel transmission term times a δ-function the full electromagnetic theory as is done in He et al[12]. A nice that picks up the appropriate angle of incidence. discussion of the derivation of transport theory from electromag- 12 netism and the conditions under which it is valid is contained in L+(z = 0; θ0, φ0) = T (ni, nt; θi, φi θ0, φ0)Li(θi, φi) an recent article by Fante[9]. The applicability of transport theory, → however, has been verified by its application to a large class of In the more general case of a rough surface, ft,s is given by a practical problems involving turbid materials, including inorganic transmission coefficient times the probability that light will refract materials such as ponds, atmospheres, snow, sand and organic in the desired direction. materials such as human skin and plant tissue[14]. The boundary conditions at the top let us formally state the Transport theory models the distribution of light in a volume contribution to reflection due to subsurface scattering in terms of by a linear integro-differential equation. the solution of the integral equation at the boundary z = 0. ∂L(~x, θ, φ) = Lr,v(θr, φr) = ft,s(θ, φ; θr, φr) L (z = 0; θ, φ) dω ∂s − Z σtL(~x, θ, φ) + σs p(~x; θ, φ; θ0, φ0)L(~x, θ0, φ0) dθ0 dφ0 − Assuming a planar surface, this integral simplifies to Z 21 Lr,v(θr, φr) = T (ni, nt; θ, φ θr, φr)L (z = 0; θ, φ) This equation is easily derived by accounting for energy balance → − within a differential volume element. It simply states that the change in radiance along a particular infinitesimal direction ds Similar reasoning allows the transmitted radiance to be deter- consists of two terms. The first term decreases the radiance due mined from the boundary conditions at the bottom boundary. to absorption and scattering. The second term accounts for light scattered in the direction of ds from all other directions. Thus, it Lt,v(θt, φt) = ft,s(θ, φ; θt, φt) L+(z = d; θ, φ) dω equals the integral over all incoming directions. Z For layered media, the assumption is made that all quantities Once again, assuming a smooth surface, only depend on z and not on x and y. This assumption is valid if the incoming illumination is reasonably constant over the region 23 Lt,v(θt, φt) = T (n2, n3; θ, φ θt, φt)L+(z = d; θ, φ) of interest. It is also roughly equivalent to saying the reflected → light emanates from the same point upon which it hits the surface. With this assumption, the above equation simplifies to Thus, the determination of the reflection functions has been reduced to the computation of L (z = 0) and L+(z = d)—the ∂L(θ, φ) solution of the one-dimensional transport− equation. cos θ = ∂z 5 Solving the Integral Equation σtL(θ, φ) + σs p(z; θ, φ; θ0, φ0) L(θ0, φ0) dθ0 dφ0 − Z There are very few cases in which integro-differential equations can be directly solved. The most famous solution is for the case of The above equation is an integro-differential equation. It can isotropic scattering and was derived by Chandrasekhar[7, p. 124]. be converted to an equivalent double integral equation, whose Even for this simple phase function the solution is anisotropic. solution is the same as the original integro-differential equation. The classic way to solve such an equation is to write it in L(z; θ, φ) = terms of the Neumann series. Physically, this can be interpreted as expanding the solution in terms of the radiance due to an integer z0 dz z σ 00 − t cos θ dz e 0 σs(z0) p(z0;θ,φ;θ0,φ0) L(z0;θ0,φ0) dω0 0 number of scattering events. That is, 0 cos θ R ∞ R R (i) This is the basis of most current approaches to volume rendering. L = L i=0 The 1-dimensional linear transport equation must also satisfy X certain boundary conditions. This is most easily seen by consid- (0) (1) ering the forward and the backward radiance separately. where L is the direct radiance assuming no scattering, L is the radiance due to a single scattering event, and L(i) is the radiance L(θ, φ) = L+(θ, φ) + L (π θ, φ) due to i scattering events. Similar equations apply to the forward − − (i) (i) and backward radiances, L+ and L . Where L is energy propagating in the positive z direction, and − + The radiance due to the i scattering events can be written using L in the negative direction. Note that L is defined to be a − − the following recurrence. function of of π θ, the angle between the backward direction − of propagation and the negative z axis. It is important to re- (i+1) member this convention when using formulas involving backward L (z; θ, φ) = z0 dz radiances. z σ 00 − t cos θ (i) dz e 0 σs(z0) p(z0;θ,φ;θ0,φ0 ) L (z0 ;θ0,φ0) dω0 0 At the top boundary the forward radiance is related to the inci- 0 cos θ R dent radiance. R R This is the basis for most iterative approaches for numerically L (z = 0; θ0, φ0) = f (θ , φ ; θ0, φ0)L (θ , φ ) dω + t,s i i i i i i calculating transport quantities. Z

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Figure 5: Solutions for fr and ft. In the left column is the surface specular reflection and in the middle is the subsurface reflection and transmission. On the right is the sum of surface Figure 4: Solutions for f (1) and f (1) for different values of g and and subsurface modulated by the Fresnel coefficients. From top r,v t,v to bottom the angle of incidence increases from 10 to 40 to 65 τd. From left to right the phase function shifts from predominately backward scattering (g = 0.3) to isotropic scattering (g = 0.0) degrees. to forward scattering (g =− 0.6). From top to bottom the optical depth of the layer increases from 0.5 to 1.0 to 2.0. A special case of this equation is Seeliger’s Law, the first attempt to model diffuse reflection from first principles[25]. Seel- 5.1 First-Order Approximation iger’s Law can be derived by assuming a semi-infinite layer (τd = ) and ignoring Fresnel effects. Another classic result in radiative transport, also derived by Chan- ∞ cos θi drasekhar[7], is the analytic solution to the integral equation as- Lr,v(θr, φr) = Li(θi, φi) suming only a single scattering event. As mentioned previously, cos θi + cos θr this is equivalent to the method described by Blinn but derived At the boundary z = d, the forward scattered radiance is given using a completely different technique [2]. by

The 0th-order solution assumes that light is attenuated by the (1) scattering and absorption, but not scattered. The attenuated inci- Lt,v(θt, φt) = cos θ dent light is called the reduced intensity and equals W T 12T 23p θ ,φ θ ,φ i e τd / cos θi e τd/ cos θt L θ ,φ ( t t; i i) cos θ cos θ ( − − ) i( i i) i− t − (0) τ/ cos θ L+ (z) = L+(z = 0)e− For cos θ = cos θ , the singular factors can be avoided by using Here, t i z L’Hospital’s rule, yielding τ(z) = σt dz (1) 12 23 τd τd/ cos θt 0 Lt,v(θt, φt) = WT T p(θt, φt; θt, φt) e− Li(θt, φt) Z cos θt is called the optical depth. If σt is constant, then τd = σtd. Using the boundary conditions for incident and reflected light, Figure 4 shows fr,v and ft,v for various values of g and d. and also rewriting the above equation in terms of the angles of Figure 5 shows the surface and subsurface components of the incidence and reflection, we arrive at the following formula for reflection model for various angles of incidence. These reflection the 0th-order transmitted intensity and transmission distribution functions have several interesting properties: (0) 12 23 τ L (θ , φ ) = T T e− d L (θ , φ ) t,v t t i i i 1. The reflection steadily increases as the layer becomes thicker; By substituting the 0th-order solution, or reduced intensity, into in contrast, the transmission due to scattering increases to a the integral equation, the 1st-order solutions for forward and back- point, then begins to decrease because of further scattering ward scattering can be calculated. The details of this calculation events. are described in Chandrasekhar and Ishimura and there is no need 2. Subsurface reflection and transmission can be predominately to repeat them here. backward or forward depending on the phase function. Using the boundary conditions for incident and reflected light, 3. As the angle of incidence becomes more glancing, the surface and also rewriting in terms of the angles of incidence and re- scattering tends to dominate, causing both the reflection and flection, we arrive at the following formula for the backscattered the transmission due to subsurface scattering to decrease. radiance: 4. Due to the Fresnel effect, the reflection goes to zero at the (1) Lr,v(θr, φr) = horizons. Also, the reflection function appears “flattened” 12 21 cos θi τ (1/ cos θ +1/ cos θr ) relative to a hemicircle. Thus, reflection for near normal WT T p(π θr ,φr ;θi,φi) (1 e− d i )Li(θi,φi) − cos θi+cos θr − angles of incidence varies less than Lambert’s Law predicts. 5. The distributions vary as a function of reflection direction. This general formula shows that the backscattered light intensity Lambert’s Law predicts a constant reflectance in all direc- depends on the Fresnel transmission coefficients, the albedo, the tions (which would be drawn as a hemicircle in these dia- layer depth, and the backward part of the scattering phase function. grams).

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1 Initialize: A particle enters the layer at the origin. Initialize p~ to the origin and the direction ~s to the direction at which the ray enters the layer. Set the weight w = 1. 999999999999 2 Events: Repeat the following steps until the ray weight drops below 999999999999 some threshold or the ray exits the layer. 999999999999 2A Step: First, estimate the distance to the next interaction: 999999999999 log r d = 999999999999 − σt 999999999999 Where r in this and the following formulas is a uniformly 999999999999 distributed random number between 0 and 1. Then, com- 999999999999 pute the new position: Figure 6: Determining first-order999999999999 solutions for multiple layers. On the left, the contribution to the first order solution for a single p~ = ~p + d ~s layer. One the right, the contribution to the first order solution And, finally set the particle weight to due to reflectance off a single layer. σs w = w σ + σ The above formulas can be used to generate first-order solutions s a for multiple layers. (This is shown diagrammatically in Figure 6.) Note: If d causes the particle to leave the layer, break from The total first-order scattering will be the sum of the first-order the repeat loop and adjust the weight using the distance to scattering from each layer, weighted by the percentage of light the boundary. making it to the layer and returning from the layer. The percentage 2B Scatter: First, estimate the cosine of the scattering angle for of light making it to the layer is the product of the 0th-order the Henyey-Greenstein phase function using the following transmission functions (or reduced intensity) for a path through formula. the layers above the reflecting layer. Similarly, the percentage of 1 1 g2 cos j = (1 + g2 ( − )2) light leaving the entire layer after reflection is equal to the product 2g − 1 g + 2gr of the 0th-order transmission functions for the path taken on the | | − way out. Note that across each boundary the light may refract, and and cos φ and sin φ with φ = 2πr. Then, compute the new thus change direction and be attenuated by the Fresnel coefficient, direction: but this is easy to handle. The process simplifies, of course, if (~s.x cos φ cos θ ~s.y sin φ)/ sin θ each layer has the same index of refraction, since no reflection ~t = (~s.y cos φ cos θ −+ ~s.x sin φ)/ sin θ or change of direction occurs between layers. Given the above sin θ ! formulas it is very easy to construct a procedure to perform this calculation and we will make use of it in the results section. ~s = ~s cos j + ~t sin j The above formula can also be generalized to include reflection Here, cos θ = ~s.z and sin θ = 1 ~s.z2. Note: Care from a boundary between layers. In many situations reflection can must be taken if sin θ = 0. − only occur from the bottom layer. In this case, we add a single p term accounting for the reduced intensity to reach the lower bound- 3 Score: Divide the sphere into regions of equal solid angle and add ary, and also weight the returning light from that boundary. Such the weight of the particle to the weight associated with the bin in which it is contained. a model is commonly employed to model the reflection of light from a pool of water[15], and has been employed by Nishita and Nakamae[22]. Further generalizations of this type are described Figure 7: Basic Monte Carlo algorithm for layered media in Ishimura[14, p. 172]. resent scattering up to some order. Note that when the albedo 6 Multiple Scattering is high, implying that σs >> σa, the first order term is only a small percentage of the total reflectance. However, as the albedo The above process of substituting the ith-order solution and then decreases, corresponding to greater absorption, a few low-order computing the integral to arrive at the (i+1)th-order solution can terms accurately approximate the reflectance. This effect can be be repeated, but is very laborious. Note that subsequent integrals explained by recalling that each term in the Neumann series rep- now involve angular distributions, because, although the input ra- resenting the reflection is on the order of W i, and since W is diance is non-zero in only a single direction, the scattered radi- always less than one, the magnitude of higher-order terms quickly ance essentially comes from the directional properties of the phase goes to zero. function. Thus, this approach to solving the system analytically We have also computed the BRDF as a function of the angle of quickly becomes intractable. reflection using our Monte Carlo algorithm for the same configu- We have implemented a Monte-Carlo algorithm for computing ration as described in the last experiment. The results are shown light transport in layered media. This algorithm is described in in Figure 9. Recall that the 1st-order reflection due to a semi- Figure 7. A thorough discussion of the application of Monte Carlo infinite media is given by Seeliger’s Law: cos θi/(cos θi + cos θr). algorithms for layered media is discussed in the book [21], and The computed 1st-order BRDF matches the theoretical result quite the techniques we are using are quite standard. well. In this figure we also plot the total BRDF due to any num- To investigate the effects of multiple scattering terms, we sim- ber of scattering events, and the difference between the total and ulated a semi-infinite turbid media with different albedos. The re- the 1st-order BRDF. Note as in the previous experiment when the flectance was computed and when the particles returning from the albedo W is small, the BRDF is closely approximated by the 1st- media are scored, we keep track of how many scattering events order term. However, note that the shape of the reflection function they underwent. Figure 8 shows the results of this experiment. is also largely determined by the shape of the 1st-order reflection, The top curve is the total reflectance, and the lower curves rep- which in turn is largely determined by the phase function. Fur-

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1.0 0.10 0.10 0.9 0.08 0.08 0.8 0.06 0.06 fr fr 0.7 0.04 0.04 0.6 0.02 0.02 0.5 0.00 0.00 0.4

Reflectance 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.3 cos r cos r 0.2 Figure 9: Graphs of the BRDF (fr) as a function of the angle 0.1 of reflection for a semi-infinite slab with different albedos (on the 0.0 left W = 0.4 and on the right W = 0.8) and an angle of inci- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 dence of 45◦. The solid line is the theoretical BRDF as given by Seeliger’s Law (the superimposed dashed line is the computed Albedo (1-W) 1st-order BRDF showing a good match). The top dashed curve Figure 8: A plot of reflectance versus albedo for a semi-infinite is the total computed BRDF; The bottom dotted curve is the dif- media. The top curve is the total reflectance (the total radiant ference between the total BRDF due to multiple scattering events energy per unit area reflected divided by the incident irradiance). and the 1st-order BRDF. The bottom curve is the reflectance assuming only a single scattering event. Moving upward is a sequence of curves consisting Property Epidermis Dermis Pigment Blood of additional terms corresponding to a single additional scatter- n 1.37-1.5 1.37-1.5 1.37-1.5 1.37-1.5 ing event. The first 10 terms in the solution are shown; In our 1 σa [mm− ] 3.8 0.3 32.6 simulations, we recorded terms involving thousands of scattering 1 σs [mm− ] 50.0 21.7 0.96 events. d [mm] 0.001-0.15 1-4 g 0.79 0.81 .79 .0 ther, observe that the difference between the 1st-order solution and the full solution is approximately independent of the angle of reflection. Thus, the sum of the higher order terms roughly obeys Table 2: Two Layer Skin Model Properties. Pigment coefficients Lambert’s Law. For this reason it is often convenient to divide are mixed with epidermal coefficients to compute the properties the subsurface reflection into two terms: of the outer layer. Blood coefficients are mixed with dermal co- (1) m efficients to compute the properties of the inner layer. Lr,v(θr, φr) = L (θr, φr) + L where Lm is constant and represents the sum of all the multiple gestive of the power of this approach; we do not claim to have an scattering terms. experimentally validated model. Finally, we have begun preliminary experiments where we in- corporate a Monte Carlo subsurface ray tracer within a standard 7.1 Skin ray tracer. When the global ray tracer calls the subsurface ray tracer it attempts to estimate the BRDF and BTDF to a particular Human skin can be modeled as two layers with almost homo- light source. This is done by biasing the Monte Carlo procedure geneous properties. Both layers are assumed to have the same to estimate the energy transported to the light. A simple method refractive index but a different density of randomly distributed to do this is to send a ray to the light at each scattering event, as absorbers and scatterers. The outer epidermis essentially consists described in Carter and Cashwell[6]. This ray must be weighted of randomly sized tissue particles and imbedded pigment parti- by the phase function and the attenuation caused by the traversal cles containing melanin. The pigment particles act as strongly through the media on the way to the light. If the albedo is less than wavelength dependent absorbers causing a brown/black coloration 1, then only a few scattering events are important, and thus the as their density increases. The inner dermis is considered to be subsurface ray tracer consumes very little time on average (the a composition of weakly absorbing and strongly scattering tissue cost is proportional the the mean number of scattering events). material and of blood which scatters light isotropically and has Also, since the subsurface ray tracer does not consider the global strong absorption for the green and blue parts of the spectrum. environment when tracing its rays, the cost of subsurface Monte Experimental evidence also supports the hypothesis that light scat- Carlo simulation at every shading calculation is relatively low. tering in the skin is anisotropic with significant forward scattering. The advantage of this approach is that the BRDF’s do not have A comprehensive study of optical properties of human skin can to precomputed, and so if material parameters are varying across be found in van Gemert et al.[27]. The values chosen for our test the surface, the correct answer is still estimated correctly at each pictures are given in Table 2. We also add a thin outer layer of point. oil that reflects light using the Torrance-Sparrow model of rough surfaces. 7 Results A head data set was acquired using a medical MRI scanner. Unfortunately, the ears and the chin were clipped in the process, The subsurface scattering models developed in this paper has been but enough of the head is visible to test our shading models. A tested on two common natural surfaces: human skin and plant volume ray tracer was adapted to output the position and normal leaves. The goal of these experiments are twofold: First, to vector of the skin layer for each pixel into a file, and this input compare our anisotropic diffuse reflection model with Lambertian was used to evaluate the shading models described in this paper. shading. Second, to attempt to simulate the optical appearance The influence of the various factors appearing in the subsur- from measured parameters. Our experiments are meant to be sug- face reflection formula are shown on Plate 1. These pictures are

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Plate 3: Dark complexion controlled by setting the concentration of melanin. On the left are images with just subsurface scattering. On the right, an specular surface term is added to simulate an oily coat. In these pictures g = .65. Plate 1.

Plate 2. not shaded in the conventional way. In particular, a Lambertian shading model would yield a constant image. The first picture (upper left) shows the influence of the Fresnel factors. Observe Plate 4: Human face with variation in subsurface blood concentration, an oily outer layer and Gaussian variation in parameters that the intensity is almost flat, but strongly attenuated for glancing incident and viewing angles. The second picture (upper middle) to create the “freckles.” shows the action of Seeliger’s Law alone. Seeliger’s Law leads to very little variation in shading, which makes the surface appear notice a much smoother “silk-like” appearance. The right column even more chalky or dusty. The third picture (upper right) demon- gives the relative difference of both models, red indicates more strates the action of the factor accounting for the finite layer depth reflection from the new model, and blue vice versa. giving only weak enhancements for glancing angles. This is a mi- To illustrate the degrees of freedom of the model, we rendered nor effect. The fourth picture (lower left) shows the influence of several faces with their parameters controlled by texture maps. the Henyey-Greenstein scattering phase function for small back- One texture map controls the relative concentration of blood in the ward scattering (g = .25) and the fifth picture (lower middle) dermis; another texture map controls the concentration of melanin − shows the effect of large forward scattering (g = .75). The result in the epidermal layer. These faces are shown in Plates 3 and 4. To is strong enhancement of glancing reflection for low angles of in- create a dark complexion we modulate the percentage of pigment cidence and viewing, assuming they are properly aligned. The last in the otherwise transparent epidermis. This creates a dark brown picture (lower right) shows the superposition of these four factors appearance due to the strong absorption of melanin (in this case we with g = .75 giving a complex behavior. An overall smoothing of set the absorption to .6). For the lips the epidermis is set to be very the reflection appears; the surface appears to be more “silk-like” thin such that the appearance is dominated by the reflection from (see also Plate 3). Although these effects are all subtle, their com- the dermis which has for the lips a large blood content (strong bination when controlled properly can create a wide variation in absorption for green and blue light component). The epidermis appearance. pigment part also has been varied locally with about 20% with a The appearance of the face with the new subsurface reflection Gaussian process. This allows us to create a wide variety of skin model is compared to the Lambertian diffuse reflection model for colors, from black to suntanned to Caucasian, and from flushed to different angles of incidence in Plate 2. The left column shows burnt to relaxed. The pictures in Plate 3 also show the effect of the results for the Lambert scattering for angles 0 and 45 degrees, an additional specular term due to a thin layer of oil on the skin. and the middle column is rendered for the new model. Again, Finally, Plate 4 shows another picture created by our program.

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Waxy Layer

Epidermis

Palisade

Spongy

Epidermis

Figure 10: Typical leaf cross-section (Redrawn from [20]). Plate 6: A cluster of leaves. A series of leaf images under different simulated lighting conditions. On the left are two backlit images, on the right, front lit.

to account for subsurface reflection and transmission. When a ray encounters a leaf, the BRDF and BTDF are evaluated for direct illumination from light sources. Shadow rays are cast to the light source, and if the ray stabs any other leaves the light intensity is attenuated by the 0th-order transmission function through each leaf. Plate 6 shows a picture of a cluster of leaves with the sun in different positions. Note that the reflection from leaves is largely determined by specular reflection due to the waxy cuticle; there is very little diffuse reflection and hence when the light source is on the same side of the leaf as the viewer, the leaf is quite dark. Plate 5: Leaf Model. On the left is the albedo image and on the The transmission term, however, can be quite large, and therefore right is a thickness image (white indicates thick) the leaves may actually be brighter when the are illuminated from behind. Note also that the increased thickness of the veins cause This picture took approximately 20 seconds to render on a Silicon dark shadows to be cast on other leaves. The veins also appear Graphics Personal Iris. dark when the leaf is back lit because they absorb more light, and bright when the leaf is front lit because their increased thickness 7.2 Leaves causes more light to be reflected. Figure 10 shows an idealized leaf in cross-section. The leaf is 8 Summary and Discussion composed of several layers of cells. On the top and bottom are epidermal cells with a thin smooth, waxy cuticular outer layer. The We have presented a reflectance model consisting of two terms: waxy cuticular layer is largely responsible for specularly reflected the standard surface reflectance and a new subsurface reflectance light. Below the upper epidermal cells there are a series of long due to backscattering in a layered turbid media. This model is palisaide cells which are highly absorbing due to the numerous applicable to biological and inorganic materials with low indices chloroplasts contained within them. Below the palisaide cells are of refraction, because their translucent nature implies that a high a loosely packed layer of irregularly shaped spongy cells. The percentage of the incident light enters the material, and so the spaces between the spongy cells are filled with air, which causes subsurface reflection is quite large. This model incorporates di- them to scatter light. Both the palisaide and the spongy cells are rectional scattering within the layer, so the resulting subsurface quite large (approximately 20 µm) compared to the wavelength, so reflection is not isotropic. This model can be interpreted as a the- their scattering phase function is forward directed. Furthermore, oretical model of diffuse reflectance. Thus, this model predicts the cells are high in water content, so the index of refraction of a directionally varying diffuse reflection, in contrast to Lambert’s the leaf is approximately equal to that of water—1.33. A typical Law. However, if multiple scattering contributes significantly to leaf is .5 to 1 mm thick, with an optical depth of 5 to 10. the reflection, then the higher scattering terms contribute to a re- To test our model on a leaf, we constructed a leaf model us- flection function with roughly the same shape. ing the technique described in Bloomenthal[3]. Although spectral As in any model, our model makes many assumptions. The two transmission and reflectance curves are available for leaves[29], most important are that the physical optics may be approximated we have set the color of the leaf from an image acquired from a with transport theory, and that the material can be abstracted into digital scanner. An albedo image is texture mapped onto a series layered, turbid media with macroscopic scattering and absorption of simply-shaped, bent polygons to create the leaf. Where the tex- properties. An “exact” model of biological tissues would explicitly ture map is transparent the polygon is considered transparent and model individual cells, organelles and so on, in considerably more the leaf is not visible. We also modulate the thickness of the leaf detail. The Monte-Carlo algorithm for simulating reflection by with a thickness map drawn on top of the original leaf image. The Westin et al.[28] is an example of such an approach. Although texture maps we used are shown in Plate 5. The waxy cuticle is such an approach may seem more accurate, often the experimental modeled using a rough specular surface with a specular exponent data needed to describe the arrangements of these structures is of 10. The interior of the leaf is modeled as a single homogeneous simply not available, and so in the end the results may be difficult layer with an optical depth of 5 and a mean scattering cosine of to validate. An advantage of the transport theory approach is that .3[20]. the parameters of the model often may be directly extracted from Pictures were generated by modifying a conventional ray tracer experimental data.

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A legitimate criticism of our work is that we did not directly [9] Fante, R. Relationship between Radiative Transport The- compare the predictions of our model with experiment. The pre- ory and Maxwell’s Equations in Dielectric Media. J. Opt. dictions of our model and the influence of measured material pa- Soc. Am. 71, 4 (April 1981), 460–468. rameters should be checked carefully. However, we believe that [10] Grawboski, L. Astrophysics J. 39 (1914), 299. this model has many applications in computer graphics even if it does not perfectly predict measured reflection functions. The [11] Hanrahan, P. From Radiometry to the Rendering Equa- metaphor of layered surfaces is very easy for users to understand tion. SIGGRAPH Course Notes: An Introduction to Radiosity because is a natural way to describe phenomenologically the ap- (1992). pearance of many materials. It also fits easily into most rendering [12] He, X. D., Torrance, K. E., Sillion, F. X., and systems and can be implemented efficiently. Greenberg, D. P. A Comprehensive Physical Model Finally, transport theory is a heuristic theory based on abstract- for Light Reflection. Computer Graphics 25, 4 (July 1991), ing microscopic parameters into statistical averages. Transport 175–186. theory is also the basis of the rendering equation, which is widely [13] Henyey, L. G., and Greenstein, J. L. Diffuse radia- viewed as the correct theoretical framework for global illumination in the galaxy. Astrophysics J. 93 (1941), 70–83. tion calculations. In this paper we propose to model surface re- Ishimura, A. flection from layered surfaces with transport theory. Thus, when [14] Wave Propagation and Scattering in Random our reflectance model for layered surfaces is incorporated into a Media. Academic Press, New York, 1978. ray tracer, there is a hierarchy of transport calculations being per- [15] Jerlov, N. G. Optical Oceanography. Elsevier, Amster- formed. Within this hierarchy, the lower level transport equation dam, 1968. computes the reflectance for the higher level transport equation. [16] Kajiya, J. Radiometry and Photometry for Computer When performing this calculation, the lower level transport equa- Graphics. SIGGRAPH Course Notes: State of the Art in Im- tion uses as its initial conditions the values from the higher level age Synthesis (1990). transport solution. Thus the two levels are coupled in a very sim- Kajiya, J. ple way. In fact, it is possible to reformulate transport theory [17] Anisotropic Reflection Models. Computer entirely in terms of reflection functions, the result is an integral Graphics 19, 3 (July 1985), 15–22. equation for the reflection function itself; in this formulation the [18] Kortum, G. Reflectance Spectroscopy. Springer-Verlag, radiance does not appear at all. Coupling transport equations at Berlin, 1969. different levels of detail in this manner is a promising approach [19] Krueger, W. The Application of Transport Theory to the to tackling the problem of constructing representations with many Visualization of 3-D Scalar Fields. Computers in Physics 5 different levels of detail as proposed by Kajiya[17]. (April 1991), 397–406. [20] Ma, Q., Ishimura, A., Phu, P., and Kuga, Y. Trans- 9 Acknowledgements mission, Reflection and Depolarization of an Optical Wave We would like to thank Craig Kolb for his help with RayShade For a Single Leaf. IEEE Transactions on Geoscience and Remote Sensing 28, 5 (September 1990), 865–872. and the leaf pictures. We would also like to thank David Laur for his help with the color plates. This research was partially [21] Marchuk, G., Mikhailov, G., Nazaraliev, M., supported by Apple, Silicon Graphics Computer Systems, David Darbinjan, R., Kargin, B., and Elepov, B. The Sarnoff Research Center, and the National Science Foundation Monte Carlo Methods in Atmospheric Optics. Springer Ver- (CCR 9207966). lag, Berlin, 1980. 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